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Presentation Embedding--Another Case of Stumbling Progress EMBEDDING: ANOTHER CASE OF STUMBLING PROGRESS Jens Høyrup [email protected] http://ruc.dk/~jensh/ Contribution to the workshop Mathematics in the Renaissance Language, Methods, and Practices ETSEIB, Barcelona, 23 January 2015 1 Approaches to algebraic symbolism Georg Nesselmann, Algebra der Griechen (1842): a classification of algebra types into three groups: rhetorical, syncopated, and symbolic. In “rhetorical algebra”, everything in the calculation is explained in full words. “Syncopated algebra” uses standard abbreviations for certain recurrent concepts and operations, while “its exposition remains essentially rhetorical”. In “Symbolic algebra”, “all forms and operations that appear are represented in a fully developed language of signs that is completely independent of the oral exposition”. 2 Though characterized as “stages” (Stufen), clear from his examples that this division into chronological stages is at most meant locally, not as steps of universal history. The rhetorical stage is represented by Iamblichos, “all so far known Arabic and Persian algebraists”, and all Christian-European writers on algebra until Regiomontanus. 3 Diophantos,and the later Europeans until well into the 17th century are classified as syncopated, although already Viète has sown the seeds of modern algebra in his writings, which however only sprouted some time after him. In the following pages, Oughtred, Descartes, Harriot and Wallis are mentioned as founders of symbolic algebra. However, we Europeans since the 17th century are not the first to have attained [the symbolic] level; indeed, the Indian mathematicians anticipate us in this domain by many centuries. 4 The typology is still in common use by historians who refuse to see every use of algebraic abbreviations as a “symbolism”, For the sane reason, it is often criticized – obvious mistakes from the 1840s would have been forgotten long ago. (In the following, I shall formulate some supplementary objections) 5 However, the central observation about what characterizes the symbolic level has mostly been overlooked – namely that symbolization allows operations directly on the level of the symbols, without any recourse to thought through spoken or internalized language. In Nesselmann’s own words We may execute an algebraic calculation from the beginning to the end in fully intelligible way without using one written word, and at least in simpler calculations we only now and then insert a conjunction between the formulae so as to spare the reader the labour of searching and reading back by indicating the connection between the formula and what precedes and what follows. 6 That is what we do when we reduce an equation by additions, divisions, differentiations, etc. We can of course speak about the operations we perform, just as we may speak about the operations we perform when changing the tyre on a bicycle or preparing a sauce; but in all three cases the operations themselves are outside language. 7 As illustration, two instances of incipient symbolic operation – one from Diophantos, one from the Italian 14th century. In the Arithmetic, Diophantos uses abbreviations (spoken of as “signs” [σημειον]) for the unknown number (the arithmós) and its powers. The unknown arithmós itself is written ς; for the higher powers (dynamis = ς2, kybos = ς3, dynamodynamis = ς4, etc.), phonetic complements are added (ΔΥ, ΚΥ, etc.). Similarly, complements are added to the sign for the monad (“power zero”), and for numbers occurring as denominators in fractions, except in the compact writing of fractions where“ 5 ” means16 . 16 5 8 Addition is implied by juxtaposition, subtraction and subtractivity are denoted by the abbreviation (λειψις, “missing” etc.). Only one sign occasionally serves direct operation: the designation of the “part denominated by” n (better, since n is not always integer, the reciprocal of n); In III.xi a number which was posited to be ς× is stated immediately to be“41 ” (77 ) when ς itself turns out to be“77 ” . 77 41 41 This would hardly have been possible if Diophantos had not known at the level of symbols (and supposed his reader to recognize) that ς× × ς p × q ( ) = , and that “q ” =“p ” . This may be the only instance of genuine symbolic operation. 9 Next: a Tuscan Trattato dell’alcibra amuchabile, a compound in three parts from c. 1365. In the third part the request to divide 100 first by a “quantity” and then by the “quantity” plus five. The sum of the quotients is told to be 20. So, you should first divide 100 by a cosa (“a thing”), and next by “a cosa and 5”, and join the two quotients. Similar problems (though with subtraction) are found in al- Khwa¯rizmı¯’s and Abu¯Ka¯mil’s algebras and again in Fibonacci’s Liber abbaci. 10 Al-Khwa¯rizmı¯ gives a purely numerical (but sensible) prescription for the initial, difficult steps obviously, what he did went beyond his technical vocabulary. Abu¯Ka¯mil uses a geometric diagram; and Fibonacci applies proportions. 11 The fourteenth-century treatise, however, comes close to what we would do: Now I want to show you something similar so that you may well understand this addition, and I shall say thus: I want to join 24 divided by 4 to 24 divided by 6, and you see that it should make 10. Therefore 24 write 24 divided by 4 as a fractions, from which comes /4. And posit similarly 24 divided by 6 as a fractions. Now multiply in cross, that is, 6 times 24, it makes 144; and now multiply 4 times 24, which is above the 6, it makes 96, join it with 144, it makes 240. Now multiply that which is below the strokes, that is, 4 times 6, it makes 25. Now you should divide 240 by 240, from which 10 should result. [...] 12 Then follows the application: Now let us return to our problem. Let us take 100 divided by a cosa and 100 divided by a cosa and 5 more, and therefore posit these two divisions as if they were fractions, as you see hereby. And now multiply in cross as we did before, that is, 100 times a cosa, which makes 100 cose. And now multiply along the other diagonal, that is, 100 times a cosa and five, it makes 100 cose and 500 in number; join with 100 cose, you get 200 cose and 500 in number. Now multiply that which is below the strokes, one with the other, it makes a censo and 5 cose more. Now multiply the results, that is, 20 against a censo and 5 cose more, it makes 20 censi and 100 things more, which quantity equals 200 things and 500 in number. ... 13 This text, we see, is purely rhetorical – everything is written out in full words. On the other hand, the solution proceeds by means of formal operations, in a way we are accustomed to in symbolic algebra; rhetorically expressed polynomials are dealt with as if they were the numbers of normal fraction arithmetic. We may say that the lexicon of the text is rhetorical, but its syntax (in part) symbolic. 14 Characteristic for this syntax is the phenomenon of embedding: the insertion of something possibly complex in the place of something simpler. We know the phenomenon from ordinary language making use of subordinate clauses: Igonow→ I go when it pleases me. 15 In contemporary symbolic mathematics indefinitely nested embedding is possible – for instance, in continued fractions, or in the graphically simpler expression x x x x x 1+ /2 (1 + /3 (1 + /4 (1 + /5 (1 + /6 (...)))) . In ordinary language, the same possibility is present, restricted only by pragmatic considerations of comprehensibility “This is the man all tattered and torn / That kissed the maiden all forlorn / That milked the cow with the crumpled horn / That tossed the dog / That worried the cat / That chased the rat / That ate the cheese / That lay in the house that Jack built”. 16 We may now turn back to Nesselmann. He ascribed to the Indian mathematicians a symbolic algebra that precedes that of Europe by many centuries. 17 An example, borrowed from Bhaskara II (b. 1115) What we would express 5x+8y+7z+90=7x+9y+6z+62 is written by Bha¯skara as a scheme yâ 5 kâ 8 nî 7 rû 90 yâ 7 kâ 9 nî 6 rû 62 while our 8x3+4x2+10y2x =4x3+0x2+12y2x appears as yâ gha 8 yâ va 4 kâ va yâ.bhâ 10 yâ gha 4 yâ va 0 kâ va yâ.bha 12 18 Datta and Singh quote David Eugene Smith for the stance that this notation is “in one respect [...] the best that has ever been suggested”, namely because it “shows at a glance the similar terms one above the other, and permits of easy transposition”. However, the Indian schemes do not permit direct multiple embedding – for instance the replacement of yâ by a polynomial. Such an operation requires the same amount of thinking in the Indian notation as in a rhetorically expressed algebra. 19 Not impossible in either case: al-Khwa¯rizmı¯’s solution of the above-mentioned problem shows that he, or somebody who inspired him, could do something of the kind; but al-Khwa¯rizmı¯ did not have the words to explain what he was doing (as he is generally fond of), free formal operations were only made possible by the fully developed symbolism of an Euler. 20 Indian schemes allow direct operations, and in this sense they clearly constitute a symbolism, as claimed by Nesselmann. However, Smith is right that this notation is the best “in one respect” only – namely within the restricted framework of problems actually dealt with by Bha¯skara. It allows operations directly at the level of symbols, but only a limited, non-expandable set of operations.
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